 Very good. So welcome to all. Today we have in this seminar to Jacopo Carusota that is going to tell us our quantum fields in Kulver space times. Please, everyone. Okay. So Raul, thanks a lot for invitation and thanks to all of you for being here with me online. So what I'm going to tell you today is a sort of introduction on the concept of analog models and some recent developments that have been done in this field in Trento and at the BC center of the National Institute of Optics in collaboration with a few young people, very brave young people who are building their career on this field. Several of them have a knowledge in scientific origin from Prieste, namely Luca and Jamia. So I hope you will be pleased to see that they are progressing very well with scientific career. So let me start from a very general introduction about what an analog model is. And to this purpose, let's consider this sort of Gedankian geologic experiment. Basically, think of a river which on upstream flows very slowly and then at some point it increases its speed until it goes into a waterfall. So suppose you are a fish and you are swimming in this region where the flow is very slow. So depending on whether your head points downstream or upstream, you will be able to swim in the downstream direction or in the upstream direction. In particular, we know that salmons are able to swim upstream for very long distances. So instead, if you are in this region where the speed of flow is faster than the maximum speed at which the fish can swim, nothing else can happen to the fish but being dragged by the river into the waterfall. Because whatever is the direction where the head of the fish points, the fish will be dragged by the current in the downstream direction compared to the reference frame of the surrounding land. So there is a point which we call horizon after which the speed of flow is faster than the speed of swim. So there is no way that the fish can cross the horizon again. So once the fish have swam across the horizon, they cannot swim back upstream. So this point of no return is a sort of horizon as it happens in astrophysical black holes. So in black holes, whenever something crosses the horizon, there is no way it can come back in the external space. And its fate is to fall into the singularity of the black hole. So in this sense, this flowing river plays a very similar role to what happens to a black hole, to an astrophysical black hole. So this is a very classical picture. So our goal, the goal of the field of analog models is to see how much this picture extends to a quantum wall. So to move towards something more quantum is better not to focus on fish but to something more abstract, which is some sound excitation. So here we have an object which flows from the left to the right with increasing speed. And if you have a loudspeaker in this fluid, you have a loudspeaker which emits waves on the subsonic side. Basically the result is that these waves will be emitted by the source, will propagate as circular waves around the loudspeaker, but they will be able to propagate both in the downstream direction and in the upstream direction. And this is because the speed of sound is larger than the speed of flow of the fluid. On the supersonic side instead, the speed of flow is larger than the speed of sound. So the sound will not be able to propagate back towards the horizon and will be dragged in the downstream direction. So sound will not be able to travel back through the horizon and exit to the external region. So in a sense this is more similar to what happens to light in an astrophysical black hole. So if you have a loudspeaker which is placed inside the horizon of the black hole and it emits waves, let's consider light to be more precise. So you have a lamp which is inside the horizon of a black hole, it will not shine light outside the black hole. Indeed the black hole is black in this sense. Instead if the lamp is placed outside the horizon, it will emit light and you will be able to see light coming out. Possibly distorted, but you will see light. So you see that this configuration is very similar to what happens in astrophysics around an astrophysical black hole. And given that we are dealing with waves, we can think that this picture could extend to some quantized radiation field, in particular to something which could support Hawking gradation. So this is the general idea of analog models which has been put into mathematical, in a mathematical language, thanks to the thought of Bilou Rue back in the 80s. And to have a very comprehensive review of this physics, I suggest to have a look at this review paper by Stefano Liberatti and co-workers where all the theoretical picture is developed. So just to in a nutshell, for the specific case of atomic gases, atomic species, the general framework is to start from a Gospidesky equation that many of you know, and write the Gospidesky equation in the limit of lower vector excitations, which have a phononic dispersion. To write this equation, for a moving fluid in a form which closely resembles the equation for quantum fields in astrophysics. And after some manipulations that are very well summarized in this review paper, you can get to an equation which has the same form of relativistic equation for a quantum field. So basically this is the operator, the box operator in the metric, Gimunu, which is a metric which depends on the properties of the flow, namely the speed of sound, the local speed of sound and the speed of flow, and some are presented. So basically the idea is that you start from the equation of motion for the for the other parameter, GP like, and with some manipulations, you can write an equation for the condensate phase in the form of a box equation in a suitable space-time metric. So this is the basic formalism of analog models. Starting from GP, this is a classical equation. The crucial leap that was done by Bilou was to consider this equation as a quantum equation for a quantum field, which means to put a hat on top of the phi. So the phi is no longer a field, a classical field, but it's an operator. So upon this quantization, one can expect this quantum field theory on a curved space-time to give rise to some Hawking emission phenomena. So the original proposal by Bilou was that this formulation will give rise to some Hawking emission of sound waves from an acoustic block. This was the 80s, but at the time there was a crucial point that there was very difficult to have systems with a suitably low temperature, such that these physics can be observed. To be concrete, let us consider the simplest black hole geometry, which basically is a speed of flow which increases along the plus-x direction, which shows an horizon at some position, and one looks for Hawking emission from this horizon point. We know that for astrophysical black holes, the Hawking temperature is an extremely low temperature. It's something which is of the order of micro-kelding for solar mass black holes, even lower for supermassive black holes that exist in galaxies. And this temperature has to be compared to the intrinsic temperature of our space, our surrounding space, which is basically dominated by the cosmic microwave background, which is of the order of a few Kelvin. So detecting a Hawking temperature of micro-kelding on top of a cosmic microwave background of the order of a few Kelvin, it's something which is extremely difficult to do and almost impossible. On the other hand, for analog models, one has the formula for the Hawking emission, the temperature of Hawking emission of sound, which is this form, which depends on the parameters of the flow, and plugging in the numbers for natural atomic clouds, one finds something which is of the order of nano-kelding. You may think that nano-kelding is completely unachievable, but actually this is not really the case. Since the laser and magnetic cooling techniques allow to obtain temperatures of a BC, which are indeed of the order of nano-kelding, exactly the same range where one expects to have the Hawking temperature. Basically, this consideration, which was done in the early 2000 by several authors, basically gave the boost to the field of analog models based on atomic compensates, because it showed that for the first time, this physics is something that could be observed in the light. But on top of this observation, there was another crucial result that was pointed out by Roberto Balbino and Sandro Fabri, and it was that the Hawking radiation in reality consists of pairs of correlated particles. Of course, this is a statement which has not by much interest in astrophysics, because it would be hard to detect some entanglement between particles which live on the opposite sides of an horizon, of a black hole horizon. So to detect the correlation between something which is inside the horizon of a black hole and something which is outside the black hole, it's just a speculation. It has no physical meaning, because the internal part of the black hole is something which is not accessible to experiment. If you want to do an experiment inside the black hole, basically you have no way to communicate the result with the external world. So it would be not a very practical thing to do. On the other hand, doing this kind of experiment in analog is something which is completely possible. In particular, working with the correlations between the Hawking particle and the partner give rise to a very neat signal of Hawking radiation which allows us to distinguish Hawking radiation from other phenomena, emission phenomena which have no special interest, like some non-zero temperature of our scent. So to see it in the analogy of the river, one would have a sort of magic stick which via quantum effects, this H-bar, emits a pair of fish from the horizon and this fish propagate on either side of the horizon in a correlated way so that these two fish and women would catch fish in a very correlated way. So whenever the lady number one catches a fish, there is an increased probability that also lady number two will catch a fish at her position. So the correlation between the Hawking and partner particle would reflect the probability, joint probability of catching fish on either side of the horizon. Of course, this is a joke but can be put in practice in some physically observable quantity. This is the result of some old Monte Carlo that we did together with Roberto, Sandro and Serena and Alessio. Basically it's a Wigner-Monte Carlo simulation of the quantum fluctuations in an atomic condensate and among many different features that appear in the G2, which is the correlation between density at position X and position X prime, you see this tongue which is highlighted by the purple circle which is a sort of mustache-like feature which says it's a correlation between a point which is outside the horizon X small over 0 and the point inside the region which is X prime into X prime larger than 0. So we are correlating points which lie on either side of the horizon and this correlation which is small but accessible has some specific geometrical properties that can be predicted by theory and it was done in this paper by Roberto and Sandro and it was confirmed in our numerical simulations in this graph. So this numerical color plot is compared to the analytical position of the tongue and we're finding a very good agreement. So this is a feature which depends on the correlation between either side of the horizon and it's something which is very, very peculiar of Hawking gradation and it was indeed used in the lab by Jeff Steinauer at Technion in Haifa to give the first hint of analog Hawking gradation in the lab. So this plot is the first experimental image of Hawking gradation taken exactly following our proposal here and this feature that's here marked as feature number three seems to be related to this sort of seagull shape which is present in the experimental graph. Of course experimental life is much harder than our theoretical life so the signal is much weaker but if you look at this graph which is an experiment is a cut of the experimental line along an orthogonal line to the moustache you clearly see the dip which corresponds very well to our theoretical prediction in here. So this seems to be what Steinauer has done in practice. So what Jeff actually did was to take a condensate. It's a very elongated condensate and it would throw it against an optical potential created by a laser beam which creates a sort of step-like potential. So this was the critical horizon. Exactly. So the horizon was formed by a condensate which was sent on a light potential which was accelerating the condensate so that a transonic point separating a sub to a supersonic flow appears at the position of this laser beam. Then what Jeff did was to take an image of the condensate and to measure the local density at the different points in 1D. So this creates a single shot of the density and then Jeff was averaging this picture over thousands of realizations. From these thousands of realizations he was able to correlate the noise in these images and to take an image of the correlations between point X and point X prime. And this black signature is visible in this noise correlation image. Okay. This was a kind of experimental tour de force. And it was even more remarkable since the first round of experiment was done by him alone. Then in the second version, from which actually this image is taken, he had some co-workers including Muahramon Munoz-Denova, who's the first author of this second nature paper. Okay. Thank you. Good. So this was the first evidence of Hawking radiation. Then a second experimental result that Jeff claimed in the nature physics paper of 16 is some experimental evidence of entanglement between the two sides of the horizon, which would be the smoking gun of the quantum nature of Hawking radiation. Here what he did was to measure some correlators between the bosonic operators on the two sides of the horizon. And he was measuring some excess anomalous correlation versus population according to this Peresord-Echki criterion. So having green, large, and red, and delta, small, and zero, he's a signature of entanglement. So his experimental data are here with the aerobars. So you see that the aerobars seem to confirm the quantum nature of his observations. This experiment is extremely difficult because you have to average and to obtain quantities which are very sensitive to the way you treat the data. So there's been in the literature a wild debate about the significance of this data. I'm not the person who is able to make a firm claim about the statistical value of this data. What I'm doing here is just to quote the results of Jeff and to pinpoint the paper where this is discussed. I have a question about this interpretation. I mean, somehow if I understood correctly this Peresord-Echki criterion, this implies that the initial state of your evolution has to be in thermal equilibrium. Otherwise, there can be some residual entanglement due to the fact that the initial state was not an equilibrium. Yes. Is there a way of cleaning up this assumption? It is that for what I did personally, I was not able. But to have entanglement, which is really dangerous, seems to be a kind of strange thing. In a sense, mathematically, it's hard to believe that that entanglement... In a sense, mathematically you cannot exclude it, but it's hard to think of a mechanism which includes entanglement exactly in the way that would disturb you. Okay. But it's an uncontrolled assumption, which in a sense is not being validated yet. There could be some mechanism that people have not thought yet which could explain this data. In a sense, I would not put my hand on fire for this experiment for many reasons. But I think it's a beautiful enough experiment, which is worth studying in full detail, and to really look at what's going on there is extremely intriguing. What I find personally very intriguing is marked here in this yellow box, is that for the first time, we are doing quantum hydrodynamics. In this sense, we are putting a hat on top of the Navier-Stokes equation. So we are doing a quantum Navier-Stokes equation. It's a bit pushed statement mine, but it's the direction where I find that these experiments are most intriguing. So we start... Can I ask a question? Sorry. Just one second, please. I finished the sentence. So we are dealing with entangled states of a macroscopic system. And I find that this development extremely intriguing. So even if this explains a statistical problem, I don't care. They're wonderful because they stimulate investigations in this direction. So please, the person who was asking question. Yes, yes, it's me. So just to understand better the theoretical model. So in the model, so you have like an inhomogeneous speed of sound? Yes. In homogenous, how inhomogeneous? Basically the speed of sound that Jeff has is more or less of this kind. V over c increases along the plus x direction and there is a point where v equals c. So for x larger than xh, it's a supersonic for x smaller than xh, it's a subsonic. And then theoretically you usually model these things really with the inhomogeneous thing or with the step function? In a sense, in my original simulations here, it was a step function in more refined numerical calculations, but we are at the moment completing. We have a precise shape of the flow, which is the one of the expert. Okay, thank you. There is one more tricky point that is worth mentioning is that the time dependence is crucial. So here in my older simulation, I took a very model time simulator, time dependence of the system. So the black hole was created in a very mathematical way, which is physically meaningful, but not experimentally achievable. The new simulations are trying to use the experimental configuration that is reused in reality to generate velocity. Which means that also the sound velocity also depends, it's a function also of time? Exactly, that's the point. This introduces some features, which could be very disturbing, which could go in the direction of, I think it was a Marcello's question about the way that uncorrelated initial fluctuations. Because having a time dependent speed of sound could introduce some unwanted entanglement between modes, which is very dangerous. Why do you say unwanted? Basically because you change the speed of sound and you are doing a sort of quench in the system, so in the language of non-equilibrium statistical mechanics. And basically this quench results in the emission of correlated particles in opposite directions. It's like having a non relativistic dispersion, essentially. Exactly. So basically you are emitting the pairs in opposite directions and these waves create some entanglement and you have to get rid of this entanglement. Otherwise this entanglement may seed the Hawking radiation, transfer to Hawking radiation and make a mess. So this is the tricky point about this assumption. That's why I'm not putting hand on fire. Thanks. But it's exactly raising a lot of interest. So this quantum hydrodynamics is one of the most interesting directions where this field will be moving in the next years. Good. So let's move on to the new results. So all I presented is something which was theoretically known until the 2015 and the new things were explained. Now let me move to some new theoretical development that we did in Trento, starting from what Luca, Luca Giammocomelli has done. Luca did his master in Trieste together with Stefano Liberati. So it's somebody that you should, where many of you should know. So basically his PhD thesis is on super radiance and the idea is to try to understand the super radiance from my quantum optician point of view and to see how analog models can give further insight on these gravitational, fundamental gravitational processes. So the new geometries that we are considering are very similar, at least one of the prototypes of these geometries is the two-dimensional vortex, which is inspiring. So there is a combination of acoustic horizon, which is the point where the velocity, the, the infalling velocity is equal to the, to the speed of sound. And there is the ergo surface, which is the point where the total velocity equals the speed of sound. So basically the acoustic horizon is the point of no return, as it was in a normal, not hitting the black hole. The ergo surface is the point after which there exist modes with negative energy. So these are two different concepts, which play a role in the, in the, in the physics we are looking. In particular, super radiance is related, is it, is an amplified reflection by transmission of a negative energy mode and basically comes from the fact that I'm incident on the black hole with some positive energy wave. This energy gets reflected with a higher intensity and the missing energy is provided by some negative energy mode, which gets excited inside the ergo surface, which drags away the negative energy, which is needed to amplify the incident. So this is the sketch picture of super radiance and basically reproduces many of the basic results, both theoretical and experimental with surface waves on this physics. But what we did with Luca was to try to understand the super radiance from a more birthing point of view, trying to, to distillate what of this physics is due to the cylindrical geometry and what is due more to kinematic properties of modes. In particular, we focused on geometries which have a translational invariance along the horizontal coordinate, where along the vertical direction there is an arbitrary dependence of the velocity. But we also assume that the irrotationally constrained typical of black holes of atomic BC is broken by using the synthetic gate field. So how comes this? So you know from superfluid hydrodynamics that normally the velocity is proportional to the gradient of the superfluid phase. This by definition is a zero curl, so it's a irrotational velocity field, which restricts a lot of the kind of flow patterns that one is available for analog models. But in quantum fluids of atoms, or also in other kind of quantum fluids, but one has the probability, the possibility of engineering a synthetic gauge field. In the presence of a gauge field, one has the different relation between the superfluid phase and the actual speed. Because the actual speed is due to the mechanical momentum. And the mechanical momentum by the minimal coupling is not only the canonical momentum, but also contains the vector field. So the actual flow speed of the quantum fluid is equal to the gradient of the phase minus the vector potential. So in this way, provided one engineers the a field in a suitable way, one can, or can build profiles with an arbitrary y-dependence of the actual physical speed. And one removes the irrotationality constraint. So this is very interesting because it widens the the amount of analog models that one can build. So in particular, we considered the geometry, which is a translational invariant along x, not with arbitrary dependence along y. And we explored the translational invariance along x to use kx as a quantum number. And we reduce the full two-dimensional problem in this metric, where vx and c depend on y, on the Klein-Goradon equation of this form, which very much resembles the Klein-Goradon equation for a charged field in an electrostatic potential, where the mass is related to the wave vector kx of the excitation along x. And the vector, the z-th component of a vector potential, say the scalar potential, is related also to the speed of flow along x, which then is a y-dependent quantum. So with this way, one can study the bosonic Klein paradox, which is basically what happens when one has charged particles hitting a jump in the scalar potential. In particular, if the jump in the scalar potential is large enough compared to the kinetic energy of the incident particles, one can have a configuration where one has the reflection of positive norm particle into negative, no, sorry, the transmission of positive norm particle as a negative norm particle, which basically a positive charged particle transforms into a negative charged particle. And in order to conserve charge, one needs to put an extra charge in the reflected wave. This is the bosonic Klein paradox, which is the sort of transmission of modulated transitions closely related to springs. So in order to study this physics, we went back to the typical workhorse of typical workhorse of GPEs, of condensates. So we did the full two-dimensional GPE independent of vector potential and scalar potential exactly as needed. And we simulated a configuration where the incident wave is a positive norm mode, which can be transmitted as a negative norm mode, which is exactly what happens here. We have an incident norm mode, which can be transmitted as a negative norm mode there. So to do this physics, we did the GPE, you see that we have an incident wave, which hits this continuity in the vector potential, gets transmitted as a negative norm mode. And the wave vector of these fringes exactly matches the wave vector of the transmitted mode. And one has a reflected wave with an increased amplitude. You see here in green is the amplitude of the incident wave, which goes to zero when it hits the surface, the vergo surface. And the reflected wave, which is plotted in blue, in red, has a higher amplitude compared to the incident wave. And here the amplification is as large as 70%. It's really a sizable amplification. So in this way, we used very simple two-dimensional GPE simulations to study super radians from a very simple configuration of an atomic gas, which shows this continuity in the vector potential of a synthetic gauge field. So the advantage of this configuration is not only that it's amenable to GPEs, but also that one can build a scattering matrix theory, which links the ingoing and outgoing Borelubov mode with positive and negative norm. So one is able to build a quantum optical picture of the super radiant phenomenon. And this is a working process. So what happens if one considers more complicated configurations? For instance, if one has a second interface, so one has a region of flow, so a region where the vector potential is non-zero only in a finite amount of space, basically what happens is that the wave packet hits the horizon, generates a wave in between the two surface. This wave goes back and forth oscillating between the two and do as time goes on gets amplified by these continuous reflections. And this amplification, which is due to super radians, turns the super radiant effect into a dynamical instability. So the dynamical instability is signaled by the fact that the wave, the amplitude of the wave in between the two interfaces grows exponentially with time, until of course non-linear effects sets in. So this shows the dynamical instability due to super radiant effects. So this is something which is related to lasing and black hole lasing. So what transforms our super radiant scatter into instability? Basically the key ingredient for going from one region to another is the fact that the light which is generated by the super radiant scattering remains available in the system for further super radiant scattering effects. So one continues to stimulate further super radiant scattering just because the wave which is created by super radiant super radians does not leave the system. So this is a very general feature. We will come back to it in a second. So this is the... Can I ask a question? Of course. What is this? So this is the bosonic super radians, right? Yes. So what is the... Imagine instead of the Klein Gordon equation, you have Dirac equation. So there should be a fermionic analog of the super radians. What would be that fermionic analog? Yeah, actually for fermion, so that's a kind of tricky question on which I've only partial answer because with Luca we are... We were studying these before all the lockdown issues, but the main difference in the two cases is marked here. You can see here. So basically you have a conservation of norm and the norm which is relevant for the bosonic problem is a norm with a non-defined metric. So you see here, the initial amplitude at a short time is 1 because you have an incident wave. So at late times you have to sum the norm of the different modes, reminding that we are transmitting negative norm modes in the internal region. So you see that the norm of the red, which is 1.7 minus 0.7, which is the norm of the blue, give 1, which was the initial norm. So in bosonic systems, you have the u square minus v square equal 1. For fermionic, you have a plus. So there is no way of having something which exponentially explodes because the u square plus v square equal 1. That's why these instability don't play a role in fermionic systems in the normal cases. But hopefully in a few months I can give you a more detailed answer if there are cases where one can go around this kind of physical. For the moment, I've not found anything. Good. So you see there is a super radiant scattering and there is a super radiant instability. So there are two different things. So this is all the work that Luca did to understand microscopically what is the meaning of super radiant. Now let's try to apply these concepts to something which is concrete. Say the quantized vertices in a quantum fluid. So you know from astrophysics that there exist two basic concepts of instabilities in black holes. One is the black hole bomb, which is basically when hooking radiation emitted by black hole is reflected at large distances and stimulates the emission of other hooking radiation. So this is a phenomenon which has been predicted to arise for light massive particles. It goes under the name of Damou, the well-off feeling instability. And somebody has even claimed that he may have a role in the dark matter content of the universe. So I'm not a good person to discuss this physics, but I find very intriguing this possible link. Anyway, this is a phenomenon which occurs because of large distance reflection at large distance from a black hole. So there's another phenomenon which is the ergo region instability. And this basically comes when you have a stellar object which is compact enough to have an ergo region. So it's a rotating star. So it has no horizon, but it's rotating fast enough to have an ergo region. And in this case, the wave which stays in the business is the one inside the ergo region. So there is a scattering at the ergo surface. The waves which are emitted inside the ergo surface stay there because there is no horizon to absorb them. So they can provide the self-amplification of the super radiant emission and give rise to some instability. So these are the two main classes of instabilities in astrophysics. So are they present in condensed matter systems? So if one looks at the vortex in a condensate, one has a complex combination of two effects. Because on one hand, the quantized vortex give an ergo region around the vortex core because the vortex in a condensate flows in a supersonic wave. But at the same time, there is a trap at large distance which basically traps the condensate. So any standard system is in a trap potentially. So what is the real physics which plays a role in an atomic condensate? Is it an ergo region instability or is it a black hole bomb instability? So this was our big question with Luca. And we went back to the literature and we found several things. We found that, of course, multiply charge vertices are always energetically unstable. A singly charge vortex in an harmonic trap is energetically unstable, but dynamically stable. Multiply charge vertices in traps are dynamically unstable, with a very complicated dependence on stability and stability on the number of atoms in the complex. So in order to understand this physics, what we try to do with Luca was to understand what happens in systems which are especially infinite geometries. So to disentangle the physics of the black hole from the physics of the trap. So to disentangle the ergo region instability from the black hole like instability from the trap. So we took as a first step a large but finite system of size tending to infinity. And we find that for large enough radii, the system tends to a constant instability rate and the instability is present for any value of the radius. So the instability domain separated by stability domain that were observed for trap systems in small sizes are an effect of the trap. But the intrinsic physics of the black hole of the vortex, sorry, is to have an ergo region like instability. To check this statement, we also did the simulation without the trap, imposing absorbing boundary conditions at large r, which is a way to directly simulate a fluid with no trap. So what we did was to send very small ingoing wave to trigger possible instability and to see how the system evolves. And what we found was that the system evolves very fast with a strong instability rate, which whose value quantitatively matches the one found before for a finite but large system. So the vortex of multiple charge turns out to be unstable in even in an infinite geometry, which means that the instability is of a ergo region kind. Black hole bomb like effects play no role in here. And in particular, the only possible effect of large distance physics is to quench the stability. So the black hole bomb effect is not there, it has rather a stabilization effect on the system. So we have confirmed the same physics in other configurations and we have found some microscopic model of these physics just by noticing that inside the ergo surface of the rotating black hole, there is a negative energy mode which can be coupled to positive energy modes which are emitted outside by parametric scattering processes. And the coupling of these two branches of modes creates some dynamical instability. As it is well known in dynamical systems, whenever you are mixing modes with different crime norms in the language of Arnold's classical mechanics book. So if you want to have fun to understand these physics in real good and deep detail, just go to my home page, look for the quantum optics course, and in there there is an exercise which is to study the dynamics of a charged harmonic oscillator with negative mass and negative spring constant. This is a physics which exactly matches the physics of super radiant instability for erotating vortex. So you can have fun solving the exercise as my student had fun in doing in last semester. Good. So this is a very general physics, it holds for any vortex no matter the charge. So it's a mechanism which really dominates the physics of vortices in conduction. So just to go back to a very basic problem, we have many of us learned from textbooks that singly charged vortices are energetically unstable, but dynamically stable. So this is a kind of misleading statement because this is true only in a harmonic trap or in other special forms of potential. If you consider a potential which is a sort of very wide harmonic trap with a small dimple in the center, sorry, and you create a singly charged vortex in this geometry, Luca realized that this configuration can be dynamically unstable, dynamically unstable under the spiraling of the vortex out of the center. So you create a vortex in the center and this guy will start spiraling out from the center within a growing radius while emitting sound waves outside in the very large condenser. The fact that it does not exist, this effect does not exist in a harmonic condenser is a sort of coincidence due to the specially harmonic trap shape which is used in normal traps. If instead of having harmonic trap you have something which at long distances is much flatter, you get the possibility of having instabilities even for singly charged vortices. Good. So this is the results which have already been kind of strongly established. Now let me spend the last 10 minutes on two new directions that are being explored in my group in collaboration with different friends starting from one issue which is, which was not yet well studied in the analog model and it's what happens if you add some two-level emitter in a curved space time quantum filter. For instance, questions could be how is Casimir effect modified if the two atoms are no longer in a flat space time but are in a curved space time? Does it introduce any change to the Casimir force between two objects? Does the presence of a neighboring black hole modify the Casimir force between two objects? All these kinds of questions which are extremely exciting from the theoretical point of view and for the moment remain a completely speculative question because I think it's very hard to go next to Sagittarius alpha and to make a measurement of Casimir effect next to a supermassive black hole. Here analog models will be the way to experimentally investigate these interesting physics of quantum filters. So to do this together with Jamia and Alessio, of course, we found a way, an atomic physics way, which is able to obtain a two-level atom which is coupled to a Borgolubo field exactly in the same way as a two-level emitter is coupled to electromagnetic field via the coupling which is the dipole times electric field. So we want a coupling which has no charge because an atom has no charge but still its polarization can couple to electromagnetic field. So here we have an atom who has no charge so it does not normally interact to the Borgolubo field but it has a sigma x operator which drives transition from the ground to the excited state and this transition couples to the fluctuations of density in the Borgolubo field. So what could be observable for this physics but the simplest fact is that the atom can be prepared in the ground state and can be moved through the condensate at supersonic speed. We know that a neutral object does not emit light but a supersonically moving atom can emit light by being excited at the excited state and at the same time emitting a photon. I have a question on this because I'm not sure I understand if you have a finite detuning over the transition that you are driving does this correspond to somehow decay of excitations in this picture because it implies that ground and excited are not symmetric anymore. So you mean in this plot? Yes that's correct. So what one is we have we are addressing a transition between G to E with some resonant light? No but the light is not resonant. Sorry? Your light is not resonant. My light is resonant with a G to E transition and we have dressed states which are superposition of E and G but the scattering length of the two states is the same with opposite sign to the fluid. So the dressed states which are a mixture of E and G have no net interaction with the outside fluid. Okay? It's okay what you said. Then the dressing between but this is not easy I mean you know it's not easy at all. Okay okay because you need to have I mean there are not many species that with the third species they have opposite. But you know the third species can be another atom. Yeah sure sure sure it can be whatever yeah but you have a species which need to have opposite scattering length with something. Do you have some species in mind do you know if there is an atom that does this actually? I don't remember I think we discussed with Carlos yes and we found that there are some possibilities playing with fresh back resonances. Basically you play with the fact that two states can have fresh back resonance in a slightly different position. You're asking something extra that they have something different and there is a point where they are exactly opposite. Exactly but you have an external parameter where you play and you tune the two and you look at the point where the two opposite by opposite. So it does not seem to be a completely crazy request. It's difficult but it one has to really to work out the tables and one possibility could be to use ions rather than atoms as an impurity. Because then you have an easier way to move them. Okay but that's it the kind of technical detail. But then to go back to this picture you have an optical field which creates death states and the two death states are detuned by the rabbit. So the transition that we use is a transition between the minus and the plus with the detuning which is the rabbit and the dipole moment of the transition is due to the fact that the two G and E independently have interaction with the atomic cloud. That is good. So in this way you can create an atom in the gram state. It moves supersonically through the condensate and creates excitations by some quantum effect where it is excited and simultaneously emits a photon. This is an effect which is known in quantum field theory. Whenever you have an emitter which moves at a super luminal speed in a quantum field. The problem is that any reasonable quantum field has a phase speed which is relativistic. So it's very difficult to have a two level atom which moves fast enough. Here you can have an impurity which moves supersonically through a PC which seems to be a reasonable thing to do. If instead of moving linearly you do it in circular motion you can have more peculiar super radiance effects that can even give rise to some instability phenomena. And this was studied in the last work together with Jamir. I have no time to discuss it in detail. Have a look at the reference and let us know if you have some questions. Just a few minutes on the last dream which is to investigate black hole evaporation in these analog models. So the very conceptual framework is the fact that there is a big question of what happens when you throw things in a black hole. So think you are throwing your mother in law in a black hole and the question is how safe are you from she coming out of it as angry as ever because information was not lost. So you throw her in the black hole and once the black hole has evaporated you find her again. Perhaps you have time to run away but perhaps it's not clear if she's lost forever. So to put it more formal way the entanglement between the external world and the inside of the black hole is it completely scrambled by the black hole nature of the black hole and information is lost or not there is remnant of what has fallen into the black hole. So this apparently simple question is raising a lot of debate in the astrophysical community. I have no way to answer but I was very intrigued and I started thinking about what happens to an analog model after that Hawking radiation makes it evaporate. So instead of attacking the complicated initial case I work in the simple configuration of a dynamical casimir emission where I shake a cavity I let light be emitted in the cavity by dynamical casimir motion but dynamical casimir from the moving mirror and look at the friction that the wall feels by the casimir emission. So the effect was calculated a long ago but it's ridiculously small so one has to find some analog models. So our first proposal done with together with Simone Dio and Cristiano is a non-optical model of a tree atom in a cavity where the dynamical casimir emission has a very clear signature in the optical absorption of the atom but it's a kind of complicated configuration. So simultaneously to the group of our friend in Messina we consider the simple case of a cavity formed by two mirrors and one of the two mirrors is attached to a spring and we look at the mechanical damping seen by one of these mirrors. So this is something which can be done in circuit 3d, blah, blah, blah. One can study the free evolution of the mechanical oscillator and it shows some mechanical friction and there is also some some signature in the response to a monochromatic drive but the most interesting thing is that this physics has a very important quantum fluctuation effects. So the effect that the mirror, the motion of the mirror is damped in time has some consequences at the level of quantum fluctuations of its position. In particular, depending on the resonance of slightly non-resonant nature of the dynamical casimir emission, the Vigner function for the mechanic motion can have very different shapes. So to study this physics what we are presently trying to do is to set up a circuit QD phenomenon where the motion of the mirror is simulated by some LC circuit which is concatenated to the squid which terminates a Copland waveguide so that just by looking at this quantum state of this extra degree of freedom one can quantum simulate the mechanical motion of the mirror. So this LC circuit plays the role of the mirror and this Copland waveguide plays the role of the casimir case. This is completely not completely crazy because there have been experiments by some group in Sweden very soon. Good, so it's time for me to conclude. So basically I try to convince you that these cold atomic gases as well as other systems are very flexible analog models to explore gravitational phenomena. So there are many systems with many phenomena which can be investigated like Hawking radiation, super radiance, but also more fancy physics which is related to emit as a couple curve spacetime quantum field theories or back reaction effect due to the back reaction of say the Hawking emission on the black color rays. These last points are present subjects of intense investigation. We hope there will be some experiments coming out in the next future. So my feeling is that this is a field which is in a phase of very strong explosion. So stay tuned and with this I thank you for your attention. Thank you Jacobo, very nice talk. Some question? Yes, so I have a question. So about this last point the back reaction effects that you were mentioning. So again to come back to the model, I mean a way to model this kind of situation should one think again to add a time dependence in the sound velocities? This actually is a tricky point because the most tricky point at present is to find the configuration where the horizon is free enough to fill the back reaction force because normal configurations are such that the horizon is bound to some external thing. So in Jeffstein's experiment the horizon is bound to the external potential. So you don't have really the way of seeing the horizon shrinking. So one of the big questions is to find a configuration where this doesn't happen. So what we are presently doing together with Giulio Buterra, who is my former post-doc who is now has a position in Glasgow, is to start the more complicated configurations like spin or condensates where one uses the density modes of the spin or condensate as a sort of background metric and the spin degrees of freedom are sort of quantum field. Of course these dynamics has nothing to do with the relativistic Einstein equations no way but leaves the metric degrees of freedom free enough to show some evolutionary response to the back reaction. Yeah no but my question was actually more basic so even before going to experiment from a theoretical point of view the description of the black hole evaporation would correspond to what in the model to put some time dependence in the speed of velocity or what? It would be that the suppose that you have a configuration which without quantum emission would be stationary. So you would see that this configuration evolves in time under the effect of quantum fluctuations. So if you think of the mirrors here if you have no dynamical cosmic emission you have that the mirror performs harmonical oscillation mode once you add the dynamical cosmic emission of photons this motion is dumped. This perhaps is the simplest way of understanding. Okay thanks. So you see you have a mirror which is neutral so in principle it feels no coupling to electromagnetic field so it should oscillate forever in the absence of that thing. So it's not related to what you were saying before about the entanglement issue of having a time dependent sound velocity. No no no this has no no no relation with that no no there's a completely different sort of toy model where you see the Hamiltonian is this so a is the cavity field b is the mechanical oscillator so you see for the two first part the first two terms in the Hamiltonian you have harmonic motion if you have the third term you have a dynamical cosmic generation of photons or pairs of photons under the effect of a mechanical motion and this in turn gives rise to a mechanical damping on the mechanical degrees of freedom. Sorry I have a question Jacopo. Yes please. Okay so this is related to the the emitter moving in the condensate can you also realize similar experiments but just in circuit Qd in which the emitter is fixed but then you modulate the resonator in spacetime or something similar. You mean the emitter physics. Yeah. This I think sir is an extremely good question I think something could be feasible in principle so basically you change the geometry of your superconducting network in a way that your emitter which is physically at rest sees an electromagnetic environment which moves in time. Yes. I think this is definitely feasible very good very good suggestion thanks. Can I have a question? Yes sir. So suppose that for some reason the underlying metric starts to emit gravitational waves how large the feedback effects from gravitational waves or the oscillations of the metric itself do you expect to be? Okay so you let me try to understand what you said so you have a black hole but the black hole does not live in empty space it is subject to some gravitational waves and then you want to look how the Kazimir how the Hawking emission is modified by these gravitational waves. This is your example. Yes. So in a sense you could ask Jeff Steinau to add in his setup an emitter of sound waves and to let these sound waves hit the horizon while he measures the Hawking radiation. I have no idea what to show up in the Hawking radiation. I think they should yes because Hawking radiation is a crucial ingredient but the system is stationary. If you have a further time dependence I think something different can can arise. I would expect that things would be very different. All right thank you. It's something that is worth being investigated. Really the formalism could be a bit cumbersome but I don't think it would be completely unfeasible because you are somehow mixing the spatial structure of a black hole with the temporal structure of a sort of dynamical Kazimir effect where you are modulating the space time in time. Okay it's complicated but it should not be a problem for a brief period. All right. All right thank you. More questions? Well if not thank you Jacob for the nice seminar. Thank you all of you for being there for the nice questions and stimulating. Perfect. Then see you. See you. We all thank you all for being there. Bye bye everybody. Ciao! Ciao. Ciao!